What is the equation, in standard form, of a parabola that models the values in the table? x -2 0 4 f(x) 2.5 1.5 -60.5
To find the equation of a parabola given a table, we need to find the values of the equation's coefficients. Since the general equation of a parabola in standard form is given by y = ax² + bx + c, we need to find the values of a, b, and c.
Plugging in the values from the table into the equation, we get three equations:
(1) 2.5 = a(-2)² + b(-2) + c
(2) 1.5 = a(0)² + b(0) + c
(3) -60.5 = a(4)² + b(4) + c
Simplifying the equations, we have:
(1) 2.5 = 4a - 2b + c
(2) 1.5 = c
(3) -60.5 = 16a + 4b + c
Now, we have a system of three equations with three unknowns:
2.5 = 4a - 2b + c
1.5 = c
-60.5 = 16a + 4b + c
Let's work on solving this system of equations.
First, we substitute the value of c from equation (2) into equations (1) and (3):
2.5 = 4a - 2b + 1.5
-60.5 = 16a + 4b + 1.5
Simplifying further, we have:
1.0 = 4a - 2b
-62.0 = 16a + 4b
Now, let's solve these two equations:
1.0 = 4a - 2b (equation 4)
-62.0 = 16a + 4b (equation 5)
Multiplying equation 4 by 2 and equation 5 by 1, we can eliminate b:
2.0 = 8a - 4b (equation 6)
-62.0 = 16a + 4b (equation 5)
Adding equations 6 and 5, we get:
-60.0 = 24a
Dividing both sides by 24, we find:
a = -60/24
a = -5/2
Now, substitute the value of a into equation 4:
1.0 = 4(-5/2) - 2b
1.0 = -10 - 2b
2b = -10 - 1.0
2b = -11.0
b = -11/2
We have found the values of a = -5/2, b = -11/2, and c = 1.5.
Now, plug these values into the general equation of a parabola in standard form to find the equation that models the values in the table:
y = ax² + bx + c
y = (-5/2)x² + (-11/2)x + 1.5
Rearranging and multiplying through by 2 to remove fractions, we have:
2y = -5x² - 11x + 3
Hence, the equation, in standard form, of the parabola that models the values in the table is:
-5x² - 11x + 2y + 3 = 0